δ(δg) * -Closed sets in Topological Spaces

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1 δ(δg) * -Closed sets in Topological Spaces K.Meena 1, K.Sivakamasundari 2 Senior Grade Assistant Professor, Department of Mathematics, Kumaraguru College of Technology, Coimabtore ,TamilNadu, India 1 Professor, Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for Women University, Coimabtore ,TamilNadu, India 2 ABSTRACT: In this paper we introduce and study a new class of generalised closed sets called δ(δg) * -closed sets in topological spaces using δg-closed sets. Moreover we analyse the relations between δ(δg) * -closed sets and already existing various closed sets. It is independent of δg-closed sets and weaker than δg*-closed sets. The class of δ(δg) * - closed sets is properly placed between the classes of δg*-closed sets and δg # -closed sets and a chain of relations is proved as follows. r-closed π-closed δ-closed δg*-closed δ(δg)*-closed δg # -closed gδ-closed KEYWORDS:.g-closed sets, δg-closed sets, δg*-closed sets and δ(δg) * -closed sets. I. INTRODUCTION The concept of generalised closed (briefly, g-closed) sets were introduced and investigated by Norman Levine 2 in Velicko 3 introduced δ-open sets which are stronger than open sets in By combining the concepts of δ- closedness and g-closedness,julian Dontchev 4 proposed a class of generalised closed sets called δg-closed set in 1996.Sudha.R and Sivakamasundari.K 20 introduced and investigated a new concept of generalised closed sets namely δg * -closed sets in 2012.The aim of this paper is to introduce a new class of generalised closed sets called δ(δg) * -closed sets. In this paper in section II we give some preliminaries which are used to carry out our work.in section III we introduce the definition of δ(δg) * -closed sets and analyse the relations between δ(δg) * -closed sets and δ-closed sets, δg * -closed sets,gδ-closed,rg-closed,gpr-closed, δg #- closed, rwg-closed, gspr-closed, πg-closed, πgp-closed, πgsp-closed, g-closed, αg-closed, *g-closed, α g -closed, #gs-closed, δg-closed, δ g -closed and gp-closed, g*p-closed, g*s-closed and (gs)*- closed sets. Throughout this paper (X, ) represents a non empty topological space on which no separation axioms are mentioned unless otherwise specified. Definition 2.1. A subset A of (X, ) is called a 1) Regular open set if A = int(cl(a)) 2) Semi open set if A cl(int(a)). 3) -open set if A int(cl(int(a))). 4) Pre-open set if A int(cl(a)). 5) Semi pre open set if Acl(int(cl(A))) 6) π-open set if it is the finite union of regular open sets. 7) δ-open set if it is the union of regular open sets. II.PRELIMINARIES Copyright to IJIRSET

2 The complements of the above mentioned sets are called regular closed, semi closed, - closed, pre- closed, Semi pre closed, π-closed and δ-closed sets respectively. The intersection of all regular closed(resp.semi-closed,-closed, pre-closed, semi pre-closed, π-closed and δ-closed) subsets of (X, ) containing A is called the regular closure(resp.semi-closure, -closure, pre- closure, semi pre closure, π-closure, and δ-closure) of A and is denoted by rcl(a) (resp. scl(a) cl(a), pcl(a), spcl(a), πcl(a), and δcl(a) ). Definition 2.2 A subset A of a topological space (X, ) is called 1) generalized closed (briefly g-closed)2 if cl(a) U whenever A U, U is open in (X, ). 2) regular generalized closed (briefly rg-closed) 5 if cl(a) U whenever A U, U is regular open in (X, ). 3) -generalized closed (briefly g-closed) 6 if cl(a) U whenever A U, U is open in (X, ). 4) generalized pre-closed (briefly gp-closed) 7 if pcl(a) U whenever A U, U is open in (X, ). 5) generalized pre regular closed(briefly gpr-closed) 8 if pcl(a)u whenever AU,U is regular open in (X, ). 6) -generalized closed (briefly g-closed) 4 if δcl(a) U whenever A U, U is open in (X, ). 7) generalized -closed (briefly g-closed) 9 if cl(a) U whenever A U, U is -open in (X, ). # 8) δg -closed 10 if δcl(a) U whenever A U, U is -open in (X, ). -closed 11 ifcl(a) U whenever A U, U is ĝ -open in (X, ). 12 if δcl(a) U whenever A U, U is ĝ -open in (X, ). 9) ĝ 10) ĝ -closed 11) g * p -closed 13 if pcl(a) U whenever A U, U is g-open in (X, ). 12) g -closed 14 if pcl(a) U whenever A U, U is ĝ -open in (X, ). 13) # gs -closed 14 if scl(a) U whenever A U, U is *g-open in (X, ). 14) g*s-closed 15 if scl(a) U whenever A U, U is gs -open in (X, ). 15) regular weakly generalised closed(briefly,rwg-closed) 16 if cl(int(a)) U whenever A U, U is regular open in (X, ). 16) generalised semi pre regular closed(briefly,gspr-closed)17 if spcl(a) U whenever AU, U is regular open in (X, ). 17) π-generalised closed(briefly,πg-closed) 18 if cl(a) U whenever A U, U is π-open in (X, ). 18) π-generalised pre- closed(briefly,πgp-closed) 19 if pcl(a) U whenever A U, U is π-open in (X, ). 19) π-generalised semi pre-closed(briefly,πgsp-closed) 17 if spcl(a) U whenever AU,U is π-open in (X,). 20) δ-generalised star closed(briefly, δg * -closed) 20 if δcl(a) U whenever A U, U is g- open in (X, ). 21) (gs) * -closed set 21 if cl(a) U whenever A U, U is gs-open in (X, ). Remark 2.3. r-closed(open) π-closed(open) δ-closed(open) δg*-closed(open) δg-closed(open) g-closed(open). Remark 2.4. For every subset A of X, spcl(a) pcl(a) δcl(a). ( Proposition 3.2 of 22 ) Copyright to IJIRSET

3 III. δ(δg) * -CLOSED SETS Definition 3.1. A subset A of a topological space (X, ) is said to be δ(δg) * -closed set if δcl(a) U whenever A U, U is δg-open in (X, ). The class of all δ(δg) * -closed sets of (X, ) is denoted by δ(δg) * C(X, ). Proposition 3.2. Every δ-closed set is δ(δg) * -closed but not conversely. Proof: Let A be a δ- closed set and U be any δg-open set containing A. Since A is δ-closed, δcl(a)=a. Therefore δcl(a)=a U and hence A is δ(δg) * -closed. Counter Example 3.3 Let X={a,b,c}, ={,X,{a},{a,b}}. In this topology the subset {c} is δ(δg) * -closed but not δ-closed. Proposition 3.4 Every δg *- closed set is δ(δg) * -closed but not conversely. Proof: Let A be δg * -closed and U be any δg- open set containing A in X. By remark 2.3. every δg-open set is g-open and A is δg * -closed, δcl(a) U. Hence A is δ(δg) * - closed. Counter Example 3.5 Let X={a,b,c},={,X,{a,b}}, Then the subset {b,c} is δ(δg) *- closed but not δg *- closed in (X, ). Proposition 3.6 Every δ(δg) * -closed set is gδ- closed but not conversely. Proof: Let A be δ(δg) * -closed set and U be any δ- open set containing A in X. By remark 2.3. every δ-open is δgopen and A is δ(δg) * -closed, δcl(a) U. For every subset A of X, cl(a) δcl(a) and so cl(a) U and hence A is gδ-closed. Counter Example 3.7 Let X={a,b,c}, ={,X,{a},{a,b},{a,c}}. Then the subset {a} is gδ-closed but not δ(δg) * - closed in (X, ). Proposition 3.8 Every δ(δg) * -closed set is rg-closed but not conversely. Proof: Let A be δ(δg) *- closed and U be any regular open set containing A in X. By remark 2.3. every regular open is δg- open and A is δ(δg) * -closed, δcl(a) U. For every subset A of X, cl(a) δcl(a) and so we have cl(a) U and hence A is rg-closed. Counter Example 3.9 Let X={a,b,c}, ={,X,{a},{a,b},{a,c}} Then the subset {a} is rg-closed but not δ(δg) * -closed in (X, ). Proposition 3.10 Every δ(δg) * -closed set is gpr-closed but not conversely. Proof: Let A be δ(δg) *- closed set and U be any regular open set containing A in X. By remark 2.3.every regular open set is δg-open and A is δ(δg) *- closed, δcl(a) U. For every subset A of X, pcl(a) δcl(a) and so we have pcl(a) U and hence A is gpr- closed. Counter Example 3.11 Let X={a,b,c}, ={,X,{a}} Then the subset {b} is gpr-closed but not δ(δg) * - closed in (X, ).. Proposition 3.12 Every δ(δg) * -closed set is δg # -closed but not conversely. Proof: Let A be δ(δg) * -closed and U be any δ-open set containing A in X. By remark 2.3. every δ-open set is δgopen and A is δ(δg) *- closed, δcl(a) U. Hence A is δg # -closed. Counter Example 3.13 Let X={a,b,c}, ={,X,{a}}.Then the subset {a} is δg # -closed but not δ(δg) * -closed in (X, ). Proposition 3.14 Every δ(δg) * -closed set is rwg-closed but not conversely. Copyright to IJIRSET

4 Proof : Let A be δ(δg) *- closed and U be any regular open set containing A in X. By remark 2.3.every regular open set is δg-open and A is δ(δg) *- closed, δcl(a) U. As int(a) A we have cl(int(a)) cl(a) δcl(a) and hence A is rwg-closed. Counter Example 3.15 Let X={a,b,c}, ={,X,{a,b}}. Then the subset {a} is rwg -closed, but not δ(δg) *- closed in (X, ). Proposition 3.16 Every δ(δg) * -closed set is gspr-closed but not conversely. Proof: Let A be δ(δg) *- closed and U be any regular open set containing A in X. By remark 2.3.every regular open set is δg-open and A is δ(δg) *- closed, δcl(a) U. By remark 2.4. spcl(a) δcl(a) and so we have spcl(a) U and hence A is gspr closed. Counter Example 3.17 Let X={a,b,c}, ={,X,{a,b}}. Then the subset {b} is gspr-closed but not δ(δg) *- closed in (X, ). Proposition 3.18 Every δ(δg) * -closed set is πg-closed but not conversely. Proof: Let A be δ(δg) *- closed and U be any π-open set containing A in X. By remark 2.3.every π-open set is δg-open and A is δ(δg) * -closed, δcl(a) U. For every subset A of X, cl(a) δcl(a) and so we have cl(a) U and hence A is πg-closed. Counter Example 3.19 Let X={a,b,c}, ={,X,{a},{b,c}}. Then the subset {c} is πg-closed but not δ(δg) * - closed in (X, ). Proposition 3.20 Every δ(δg) * -closed set is πgp-closed but not conversely. Proof: Let A be δ(δg) * -closed and U be any π-open set containing A in X. By remark 2.3.every π-open set is δg- open and A is δ(δg) * -closed, δcl(a) U. By remark 2.4. pcl(a) δcl(a) and so we have pcl(a) U and hence A is πg-closed. Counter Example 3.21 Let X={a,b,c}, ={,X,{a}} Then the subset {a} is πgp-closed but not δ(δg) * -closed in (X, ). Proposition 3.22 Every δ(δg) * -closed set is πgsp-closed but not conversely. Proof: Let A be δ(δg) * -closed set and U be any π-open set containing A in X. By remark 2.3.every π-open set is δgopen and A is δ(δg) * -closed, δcl(a) U. By remark 2.4. spcl(a) δcl(a) and so we have spcl(a) U and hence A is πgsp-closed. Counter Example 3.23 Let X={a,b,c}, ={,X,{a},{b},{a,b}}. Then the subset {a} is πgsp-closed but not δ(δg) * - closed in (X, ). Remark 3.24 The following figure gives the dependence of δ(δg) * -closed set with eleven closed sets. gδ-closed rg-closed δ-closed δg*-closed δ(δg)*-closed πgp-closed gpr-closed δg#-closed rwg-closed gspr-closed πg-closed πgsp-closed Remark 3.25 The following counter examples show that δ(δg) * -closedness is independent from g-closedness, δg-closedness, αg-closedness, *g-closedness, α g -closedness, #gs-closedness, δ g -closedness, gp-closedness and Copyright to IJIRSET

5 g*p-closedness. ISSN: Counter Example 3.26 Let X={a,b,c}, ={,X,{a}}. In this topology the subset {b} is g-closed, δg-closed, αg-closed, *g-closed, α g -closed, #gs-closed, δ g -closed, g * -closed, gp-closed and g*p-closed but not δ(δg) * -closed. Counter Example 3.27 Let X={a,b,c}, ={,X,{a},{a,b},{a,c}} In this topology the subset {a,b} is δ(δg) * -closed but not g-closed, δg-closed, αg-closed, *g-closed, α g -closed, #gs-closed, δ g -closed, gp-closed and g*p-closed. Remark 3.28 The following Counter Example shows that δ(δg) * -closedness is independent from g*s-closedness. Counter Example 3.29 Let X={a,b,c}, ={,X,{a},{a,b}}. In this topology the subset {a,c} is δ(δg) * -closed but not g*s-closed and the subset{b} is g * s closed but not δ(δg) * -closed. Remark 3.30 The following Counter Example shows that δ(δg) * -closedness is independent from (gs)*-closedness. Counter Example 3.31 Let X={a,b,c}, ={,X,{a,b}}. In this topology the subset {b,c} is δ(δg) * -closed but not (gs)*-closed. Counter Example 3.32 Let X={a,b,c}, ={,X,{a},{b,c}}. In this topology the subset {a,c} is (gs)*-closed but not δ(δg) * -closed. Remark 3.33 From the above relations between various g-closed sets using δ-closure we get the following implications. r-closed π-closed δ-closed δg*-closed δg-closed gδ-closed δ g -closed δ(δg) * -closed δg # -closed REFERENCES 1 Levine, N. (1963), Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly., 70, Levine, N. (1970), Generalized closed sets in topology, Rend.Circ.Math.Palermo, 19, Velicko,N.V.(1968), H-closed topological spaces, Amer. Math. Soc. Transl., 78, Palaniappan N and KC Rao (1993) Regular generalized closed sets. Kyungpook Math. J 33: Dontchev,J.&Ganster.M.(1996),On δ-generalised closed sets and T 3/4 spaces,mem.fac.sci.kochi.univ.math.17, Maki H, R Devi and K Balachandran (1994) Associated topologies of generalized -closed Sets and α-closed sets and α-generalized closed sets. Mem. Fac. Sci. Kochi Univ. (Math) 15: Maki H,J.Umehara &T.Noiri (1996)Every topological space is pre-t½.mem Fac Sci Kochi Univ Ser.Alath17, Gnanambal.Y(1997)On generalized pre regularclosed sets in topological spaces.ind.j.pure.appl.math.28(3): Julian Dontchev,I.Arokiarani and Krishnan Balachandran(1997) On generalised δ-closed sets and almost weakly Hausdorff spaces. Topology Atlas Copyright to IJIRSET

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